Find the sum of the first $30$ terms in this geometric series: $-5 -4 -\dfrac{16}5...$ Choose 1 answer: Choose 1 answer: (Choice A) A $-1.24\cdot10^{20}$ (Choice B) B $ -24.97 $ (Choice C) C $-2.78$ (Choice D) D $-0.04$
Getting started We're dealing with a geometric series because each term is multiplied by $\dfrac45$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-5})$ and the number of terms $(n = {30})$ are given in the question. The common ratio $r$ is ${\dfrac45}$ because each term is multiplied by ${\dfrac45}$ to get the next term. [How did we find the common ratio r?] Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{30}}&=\dfrac{{-5}\left(1-\left({\dfrac45}\right)^{{30}}\right)}{1-\left({\dfrac45}\right)} \\\\ S_{{30}}&=-25\left(1-\left({\dfrac45}\right)^{{30}}\right)\\\\ S_{{{30}}} &\approx -24.97 \end{aligned}$ The answer $ -24.97 $